[FOM] vagueness in mathematics?

[Henning Basold]

On 17/02/17 03:22, Timothy Y. Chow wrote:
> Charlie Silver wrote:
> 
>>     To me, this is expressed simpler by asking what it means to
>>     "continue in the same way," which seems the point of Tim Chow's
>>     example.  For instance, if we begin with 1, 2, 3, and try to
>>     "continue in the same way," what stops us from this: 1, 2, 3, 1,
>>     2, 3, ?  Any finite sequence can be followed by any other. Another
>>     example would be 1, 2, 3, 101, 102, 103, 201,?
> 
> There's more to Kripkenstein than that.  There is a glib response
> available to the challenge to "continue in the same way," which is that
> you have failed to say what "continue in the same way" means.  The glib
> response continues by saying that if you were to be more specific or
> precise, e.g., by saying that the sequence is the sequence of natural
> numbers, or the sequence is the sequence "1,2,3" repeated indefinitely,
> then the vagueness would disappear.
> 
> The Kripkensteinian skeptic goes further, by insisting that *there is no
> way to specify a rule* that says what the rest of the sequence is.  For
> example, if you try to say that the rule is "1,2,3,1,2,3 repeated
> indefinitely" then I will skeptically ask, what do you mean by
> 1,2,3,1,2,3 repeated indefinitely?
> 
> Tim: Is "1,2,3,1,2,3,4" an initial segment of "1,2,3,1,2,3 repeated
> indefinitely"?
> 
> Charlie: Say what?  Of course not; didn't you hear me?  I said
> 1,2,3,1,2,3 repeated indefinitely, which means that the next number is
> 1, and the next one after that is 2, and the next one after that is 3. 
> That's the rule.
> 
> Tim: A rule?  I thought that I knew what you meant by a rule, but now
> I'm not so sure.  It seemed quite clear to me that the rule "1,2,3,1,2,3
> repeated indefinitely" meant that the next number was obviously 4.  But
> you're telling me that that "rule" dictates that the next number is 1?
> Weird.  What do you mean by a "rule" then?
> 
> The argument is not just that any finite sequence can be continued in
> infinitely many ways.  The argument is that even after you say, with as
> much precision and explicitness as the entire mathematical community can
> muster, *exactly what the rule* for the sequence is, the sequence is
> *still* not determined.  Any finite amount of natural language
> conversation, gesticulation, drawing of pictures, training in logic,
> building of computers and demonstration of their operation, holding of
> international conferences, browbeating, and foaming at the mouth will
> still fail to nail down even the simplest possible so-called "rule."
> Everyone could agree on every instance of the alleged rule that has ever
> come up in the history of mankind but it could all just be a huge lucky
> coincidence.
> 
> Tim

Dear Tim,

I agree with the failure of descriptions of sequences by means of
"continue in the same way". However, I would like to discuss your last
point further, namely that there is no way to determine a sequence by a
rule. I am pretty sure that you know all of what I am going to say, and
it may very well be that I misunderstood your email. The intention is
just to discuss a different view here as well.

The problem I see here is what we understand a sequence to be.
Apparently, these are usually conceived as a list of numbers (or other
things) that never ends, which is clearly impossible to write down. But
is this really how we work with sequences? If we look at how sequences
are used in, say, analysis, then we find that these are usually given by
writing s = (e)_{n ∈ ℕ}, where e is some finite expression that may use
n. Since e is a finite expression, we obtain a finite representation of
the sequence *associated* to s. That is to say, we can evaluate e at any
n ∈ ℕ we like. Thus we could *define* sequences to be just given by such
an expression over the index.

Another approach is that we see sequences as processes that can
compute/choose values at any particular index we like. This the approach
that Brouwer, Dummett and others took (see choice sequences). More
recently, the advent of coalgebra (and coinductive types) gives a
cleaner view of this: A sequence is seen in this world as a process that
we can ask for the first element of the sequence it represents (the
so-called head) and for the process that continues the computation after
this first element (the tail).

In this latter view, the sequences that can be expressed depend of
course of what processes we allow (c.f. lawless vs. lawlike vs.
computable). But surely, there are systems that allow us to specify a
process that represents a sequence precisely and without any ambiguity
(Stream Coalgebras; Stream Differential Equations by Jan Rutten;
Corecursion Schemes à la Hagino, Mendler, and others; Copattern Calculus
by Abel et al.; etc.).

These systems certainly do not encompass all possible sequences, since a
sequence must be finitely representable to fit into those systems.
However, this is not the point, the point is that if we view sequences
as processes that we can ask for their values instead of the "infinite
list of values" the denote, then *there are* rules that determine a
sequence uniquely, given that we work in a reasonable system of mathematics.

Kind Regards,
Henning

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